Taming the fourth dimension

WHEN the brilliant French mathematician Henri Poincaré was not actually doing maths, he liked to think about the nature of mathematical creativity. Logic, he felt, was important, but it was not enough. “It is by logic we prove,” Poincaré wrote, “it is by intuition that we invent.”

The problem, as Poincaré would have acknowledged, is that there is a disconnect between intuition and logic. Mathematicians often have intuitions, which they call conjectures or hypotheses. Many of these are remarkably resistant to logical proof. One of the most famous is the Poincaré conjecture. His conjecture is concerned with topology, the study of shapes, spaces and surfaces.

Think, for example, of the way a noose looped round the Earth might be tightened and its circumference eventually reduced to zero as it slips off at the North Pole. Try to accomplish the same trick with a noose formed through the handle of a ...

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